= Cohomology and base change
Cohomology and base change asks whether changing the base of a family commutes with taking its <sheaf cohomology>. For a scheme $X$ over a ring $A$, an $A$-algebra $B$, and a <quasi-coherent sheaf> $\mathcal F$, the natural comparison is
$$
H^q(X,\mathcal F)\otimes_A B\longrightarrow H^q(X_B,\mathcal F_B),\qquad X_B=X\times_{\operatorname{Spec}A}\operatorname{Spec}B.
$$
Here $\mathcal F_B$ is the pullback. The left side first computes cohomology and then changes scalars; the right side changes the geometric base first.
Flat base change gives an isomorphism for every $q\geq0$ when $A\to B$ is flat and $X\to\operatorname{Spec}A$ is quasi-compact and quasi-separated. On separated schemes, the mechanism is a Čech complex from a finite affine cover: flat tensor product preserves the kernels and images needed to compute its cohomology.<The Stacks project>[Tag02KH, Lemma30.5.2, including affine-base formulation and proof]{ref}
Without flatness, tensor product need not commute with cohomology. The two-term complex $k[t]\xrightarrow{t}k[t]$ in degrees zero and one has zero degree-zero cohomology, but tensoring with $k[t]/(t)$ makes the differential zero and produces a nonzero degree-zero group. This illustrates the algebraic obstruction; proper-family base-change theorems impose additional hypotheses to control it.
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